**Classification of Numbers Worksheet for Grade 11 :**

Here we are going to see some practice questions on classifying numbers.

**Natural numbers:**

Set of all numbers which is beginning with 1, 2,... is called as Natural numbers

Symbol for natural number = ℕ

**Whole numbers:**

Set of all numbers which is beginning with 0, 1, 2, 3...is called as whole numbers.

Symbol for natural number = W

**Integers:**

Integers are set of all whole numbers and their opposites. We are using number line to denote integers....-3, -2, -1, 0, 1, 2, 3....

Symbol for integers = ℤ

**Rational numbers**

In our number system we are representing the rational number in the form of fraction like a/b.

1.5 = 3/2

Symbol for rational number = ℚ

**Irrational numbers**

An irrational number is any real number which cannot be expressed as a simple fraction or rational number. Π, √2 are some examples or irrational numbers.

Symbol for rational number = R - ℚ

**Real numbers**

All numbers including positive integers, negative integers, rational numbers, irrational numbers, are called real numbers.

It is usually denoted as ℝ

**Question 1 :**

Classify each element of {√7, −1/4 , 0, 3.14, 4, 22/7} as a member of N, Q, R − Q or Z.

**Solution :**

√7 is irrational number

√7 ∈ R - ℚ

−1/4 is a rational number

−1/4 ∈ R - ℚ

0 is a integer

0 ∈ Z

3.14 is a rational number

3.14 ∈ Q

4 is a integer

4 ∈ Z

22/7 is a rational number.

22/7 ∈ Q

**Question 2 :**

Prove that √3 is an irrational number.

(Hint: Follow the method that we have used to prove √2 ∉ Q.)

**Solution :**

Let √3 be a rational number

√3 = p/q

p and q are co primes.

Co prime means the numbers which has common divisor other than 1.

p = √3 q

Taking squares on both sides, we get

p^{2} = (√3 q)^{2}

p^{2} = 3 q^{2}

q^{2 }= p^{2}/3

p is the divisor of 3.

p/3 = a

p = 3a

taking squares on both sides, we get

p^{2} = 9a^{2}

Here p^{2} = 3q^{2}

(3q^{2}) = 9a^{2}

q^{2} = 9a^{2}/3

a^{2 }= q^{2}/3

q is the factor of 3.

This contradicts out assumption. Hence √3 is irrational number.

**Question 3 :**

Are there two distinct irrational numbers such that their difference is a rational number? Justify.

**Solution :**

Let the two irrational numbers be 5 + √3 and 5 - √3.

Difference = (5 + √3) - (5 - √3)

= 5 + √3 - 5 + √3

= 2√3

Hence two distinct irrational numbers such that their difference is a rational number

**Question 4 :**

Find two irrational numbers such that their sum is a rational number. Can you find two irrational numbers whose product is a rational number.

**Solution :**

Let the two irrational numbers be 5 + √3 and 5 - √3.

Sum = (5 + √3) + (5 - √3)

= 5 + √3 + 5 - √3

= 10

Product = (5 + √3) (5 - √3)

= 5^{2} - √3^{2}

= 25 - 3

= 22

Question 5 :

Find a positive number smaller than 1/2^{1000} . Justify.

Solution :

We see

1/2^{1} > 1/2^{2} > 1/2^{3}

Hence the positive number less than 1/2^{1000 }is 1/2^{1001}

After having gone through the stuff given above, we hope that the students would have understood, "Classification of Numbers Worksheet for Grade 11"

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